Analysis on the steady Euler flows with stagnation points in an infinitely long nozzle

Abstract

Flows with stagnation points, very challenging in analysis, are interesting and important phenomena in fluids. In this paper, we not only prove the uniqueness and existence of steady flows with a stagnation set, but also obtain the regularity of the boundary of the stagnation set, which is a class of obstacle-type free boundary. First, we prove a global uniqueness theorem for solutions of the two-dimensional steady Euler system, whose horizontal velocities tend to those of shear flows at the far field upstream. Due to the appearance of stagnation points, the nonlinearity of the semilinear equation for the stream function becomes non-Lipschitz. This creates a challenging analysis problem since many classical analysis methods do not apply directly. Second, the existence of steady incompressible Euler flows is established in an infinitely long nozzle via a variational approach. A very interesting phenomenon is that the boundary of a stagnation region can be regarded as an obstacle-type free boundary, which is proved to be globally $C^1$. Finally, the existence of a stagnation set is proved as long as the nozzle is wider than the width of the nozzle upstream, where the flows tend to Euler flows with stagnation points.